Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry

Semidefinite programming (SDP) bounds for the quadratic assignment problem (QAP) were introduced in Zhao et?al. (J Comb Optim 2:71–109, 1998). Empirically, these bounds are often quite good in practice, but computationally demanding, even for relatively small instances. For QAP instances where the data matrices have large automorphism groups, these bounds can be computed more efficiently, as was shown in Klerk and Sotirov (Math Program A, 122(2), 225–246, 2010). Continuing in the same vein, we show how one may obtain stronger bounds for QAP instances where one of the data matrices has a transitive automorphism group. To illustrate our approach, we compute improved lower bounds for several instances from the QAP library QAPLIB.     

Generating Hard Test Instances with Known Optimal Solution for the Rectilinear Quadratic Assignment Problem

In this paper we consider the rectilinear version of the quadratic assignment problem (QAP). We define a class of edge-weighted graphs with nonnegatively valued bisections. For one important type of such graphs we provide a characterization of point sets on the plane for which the optimal value of the related QAP is zero. These graphs are used in the algorithms for generating rectilinear QAP instances with known provably optimal solutions. The basic algorithm of such type uses only triangles. Making a reduction from 3-dimensional matching, it is shown that the set of instances which can be generated by this algorithm is hard. The basic algorithm is extended to process graphs larger than triangles. We give implementation details of this extension and of four other variations of the basic algorithm. We compare these five and also two existing generators experimentally employing multi-start descent heuristic for the QAP as an examiner. The graphs with nonnegatively valued bisections can also be used in the construction of lower bounds on the optimal value for the rectilinear QAP.     

几种基于匈牙利算法求解二次分配问题的方法及其分析比较

二次分配问题(Quadratic assignment problem,QAP)属于NP-hard组合优化难题。二次分配问题的线性化模型和下界计算方法,是求解二次分配问题的重要途径。本文以二次分配问题的线性化模型为基础,根据现有QAP对偶上升下界计算方法中的具体操作,提出几种可行的QAP对偶上升计算新方法。最后,通过求解QA-PLIB中的部分实例,深入分析其运行结果,详细讨论了基于匈牙利算法求解二次分配问题的对偶方法中哪些操作可较大程度地提高目标函数最优解的下界增长速度,这为基于匈牙利算法求解二次分配问题的方法的改进奠定了基础。     

一种基于匈牙利算法的二次分配问题求解方法

二次分配问题(Quadratic assignment problem,QAP)属于NP-hard组合优化难题.二次分配问题的线性化及下界计算方法,是求解二次分配问题的重要途径.以Frieze-Yadegar线性化模型和Gilmore-Lawler下界为基础,详细论述了二次分配问题线性化模型的结构特征,并分析了Gilmore-Lawler下界值往往远离目标函数最优值的原因.在此基础上,提出一种基于匈牙利算法的二次分配问题对偶上升下界求解法.通过求解QAPLIB中的部分实例,说明了方法的有效和可行性.     

Contributions to the quadratic assignment problem

Tree search procedures for solving the Koopmans Beckmann quadratic assignment problem (QAP) are unable to solve any reasonable size QAP's mainly because good quality lower bounds for this problem cannot be computed.The purpose of this paper is to propose a bounding technique based on the extraction from the QAP formulation, of a large linear assignment problem (which can then be solved optimally), leaving as a residual problem as ‘small’ a QAP as possible. The solution of this residual QAP can then be bounded by a separate procedure. This 2-step method produces improved bounds as compared with those produced by the direct application of the bounding algorithms to the original QAP. In addition, a procedure is described for the a priori fixing of variables in the QAP formulation, thus reducing the number of variables in the problem.     

Ranking scalar products to improve bounds for the quadratic assignment problem

The eigenvalue bound for the quadratic assignment problem (QAP) is successively improved by considering a set of k-best scalar products, related to the QAP. An efficient procedure is proposedto find such a set of k-best scalar products. A class of QAPs is described for which this procedure in general improves existing lower bounds and at the same time generates good suboptimal solutions. The method leaves the user with a large flexibility in controlling the quality of the bound. However, since the method is sensitive to input data it should only be used in combination with other bounding rules.     

A new exact discrete linear reformulation of the quadratic assignment problem

The quadratic assignment problem (QAP) is a challenging combinatorial problem. The problem is NP-hard and in addition, it is considered practically intractable to solve large QAP instances, to proven optimality, within reasonable time limits. In this paper we present an attractive mixed integer linear programming (MILP) formulation of the QAP. We first introduce a useful non-linear formulation of the problem and then a method of how to reformulate it to a new exact, compact discrete linear model. This reformulation is efficient for QAP instances with few unique elements in the flow or distance matrices. Finally, we present optimal results, obtained with the discrete linear reformulation, for some previously unsolved instances (with the size n = 32 and 64), from the quadratic assignment problem library, QAPLIB.     

Minimum energy configurations on a toric lattice as a quadratic assignment problem

We consider three known bounds for the quadratic assignment problem (QAP): an eigenvalue, a convex quadratic programming (CQP), and a semidefinite programming (SDP) bound. Since the last two bounds were not compared directly before, we prove that the SDP bound is stronger than the CQP bound. We then apply these to improve known bounds on a discrete energy minimization problem, reformulated as a QAP, which aims to minimize the potential energy between repulsive particles on a toric grid. Thus we are able to prove optimality for several configurations of particles and grid sizes, complementing earlier results by Bouman et al. (2013). The semidefinite programs in question are too large to solve without pre-processing, and we use a symmetry reduction method by Permenter and Parrilo (2020) to make computation of the SDP bounds possible.     

Recent Advances for the Quadratic Assignment Problem with Special Emphasis on Instances that are Difficult for Meta-Heuristic Methods

This paper reports heuristic and exact solution advances for the Quadratic Assignment Problem (QAP).QAPinstances most often discussed in the literature are relatively well solved by heuristic approaches. Indeed, solutions at a fraction of one percent from the best known solution values are rapidly found by most heuristic methods. Exact methods are not able to prove optimality for these instances as soon as the problem size approaches 30 to 40. This article presents new QAP instances that are ill conditioned for many metaheuristic-based methods. However, these new instances are shown to be solved relatively well by some exact methods, since problem instances up to a size of 75 have been exactly solved.     

A Tabu Search Algorithm for the Quadratic Assignment Problem

Tabu search approach based algorithms are among the widest applied to various combinatorial optimization problems. In this paper, we propose a new version of the tabu search algorithm for the well-known problem, the quadratic assignment problem (QAP). One of the most important features of our tabu search implementation is an efficient use of mutations applied to the best solutions found so far. We tested this approach on a number of instances from the library of the QAP instances—QAPLIB. The results obtained from the experiments show that the proposed algorithm belongs to the most efficient heuristics for the QAP. The high efficiency of this algorithm is also demonstrated by the fact that the new best known solutions were found for several QAP instances.     

Bounds for the quadratic assignment problem using the bundle method

Semidefinite programming (SDP) has recently turned out to be a very powerful tool for approximating some NP-hard problems. The nature of the quadratic assignment problem (QAP) suggests SDP as a way to derive tractable relaxations. We recall some SDP relaxations of QAP and solve them approximately using a dynamic version of the bundle method. The computational results demonstrate the efficiency of the approach. Our bounds are currently among the strongest ones available for QAP. We investigate their potential for branch and bound settings by looking also at the bounds in the first levels of the branching tree.      

Improved lower bounds and exact algorithm for the capacitated arc routing problem

In the capacitated arc routing problem (CARP), a subset of the edges of an undirected graph has to be serviced at least cost by a fleet of identical vehicles in such a way that the total demand of the edges serviced by each vehicle does not exceed its capacity. This paper describes a new lower bounding method for the CARP based on a set partitioning-like formulation of the problem with additional cuts. This method uses cut-and-column generation to solve different relaxations of the problem, and a new dynamic programming method for generating routes. An exact algorithm based on the new lower bounds was also implemented to assess their effectiveness. Computational results over a large set of classical benchmark instances show that the proposed method improves most of the best known lower bounds for the open instances, and can solve several of these for the first time.     

A Boolean satisfiability approach to the resource-constrained project scheduling problem

We formulate the resource-constrained project scheduling problem as a satisfiability problem and adapt a satisfiability solver for the specific domain of the problem. Our solver is lightweight and shows good performance both in finding feasible solutions and in proving lower bounds. Our numerical tests allowed us to close several benchmark instances of the RCPSP that have never been closed before by proving tighter lower bounds and by finding better feasible solutions. Using our method we solve optimally more instances of medium and large size from the benchmark library PSPLIB and do it faster compared to any other existing solver.     

A survey for the quadratic assignment problem

The quadratic assignment problem (QAP), one of the most difficult problems in the NP-hard class, models many real-life problems in several areas such as facilities location, parallel and distributed computing, and combinatorial data analysis. Combinatorial optimization problems, such as the traveling salesman problem, maximal clique and graph partitioning can be formulated as a QAP. In this paper, we present some of the most important QAP formulations and classify them according to their mathematical sources. We also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms. We then give a detailed discussion of the progress made in both exact and heuristic solution methods, including those formulated according to metaheuristic strategies. Finally, we analyze the contributions brought about by the study of different approaches.     

A cooperative parallel tabu search algorithm for the quadratic assignment problem

In this study, we introduce a cooperative parallel tabu search algorithm (CPTS) for the quadratic assignment problem (QAP). The QAP is an NP-hard combinatorial optimization problem that is widely acknowledged to be computationally demanding. These characteristics make the QAP an ideal candidate for parallel solution techniques. CPTS is a cooperative parallel algorithm in which the processors exchange information throughout the run of the algorithm as opposed to independent concurrent search strategies that aggregate data only at the end of execution. CPTS accomplishes this cooperation by maintaining a global reference set which uses the information exchange to promote both intensification and strategic diversification in a parallel environment. This study demonstrates the benefits that may be obtained from parallel computing in terms of solution quality, computational time and algorithmic flexibility. A set of 41 test problems obtained from QAPLIB were used to analyze the quality of the CPTS algorithm. Additionally, we report results for 60 difficult new test instances. The CPTS algorithm is shown to provide good solution quality for all problems in acceptable computational times. Out of the 41 test instances obtained from QAPLIB, CPTS is shown to meet or exceed the average solution quality of many of the best sequential and parallel approaches from the literature on all but six problems, whereas no other leading method exhibits a performance that is superior to this.     

Extensive Testing of a Hybrid Genetic Algorithm for Solving Quadratic Assignment Problems

A robust search algorithm should ideally exhibit reasonable performance on a diverse and varied set of problems. In an earlier paper Lim et al. (Computational Optimization and Applications, vol. 15, no. 3, 2000), we outlined a class of hybrid genetic algorithms based on the k-gene exchange local search for solving the quadratic assignment problem (QAP). We follow up on our development of the algorithms by reporting in this paper the results of comprehensive testing of the hybrid genetic algorithms (GA) in solving QAP. Over a hundred instances of QAP benchmarks were tested using a standard set of parameters setting and the results are presented along with the results obtained using simple GA for comparisons. Results of our testing on all the benchmarks show that the hybrid GA can obtain good quality solutions of within 2.5% above the best-known solution for 98% of the instances of QAP benchmarks tested. The computation time is also reasonable. For all the instances tested, all except for one require computation time not exceeding one hour. The results will serve as a useful baseline for performance comparison against other algorithms using the QAP benchmarks as a basis for testing.     

An exact algorithm for large multiple knapsack problems

The Multiple Knapsack Problem (MKP) is the problem of assigning a subset of n items to m distinct knapsacks, such that the total profit sum of the selected items is maximized, without exceeding the capacity of each of the knapsacks. The problem has several applications in naval as well as financial management. A new exact algorithm for the MKP is presented, which is specially designed for solving large problem instances. The recursive branch-and-bound algorithm applies surrogate relaxation for deriving upper bounds, while lower bounds are obtained by splitting the surrogate solution into the m knapsacks by solving a series of Subset-sum Problems. A new separable dynamic programming algorithm is presented for the solution of Subset-sum Problems, and we also use this algorithm for tightening the capacity constraints in order to obtain better upper bounds. The developed algorithm is compared to the mtm algorithm by Martello and Toth, showing the benefits of the new approach. A surprising result is that large instances with n=100 000 items may be solved in less than a second, and the algorithm has a stable performance even for instances with coefficients in a moderately large range.     

Ant colony optimization for solving an industrial layout problem

This paper presents ACO_GLS, a hybrid ant colony optimization approach coupled with a guided local search, applied to a layout problem. ACO_GLS is applied to an industrial case, in a train maintenance facility of the French railway system (SNCF). Results show that an improvement of near 20% is achieved with respect to the actual layout. Since the problem is modeled as a quadratic assignment problem (QAP), we compared our approach with some of the best heuristics available for this problem. Experimental results show that ACO_GLS performs better for small instances, while its performance is still satisfactory for large instances.     

Biological computation of the solution to the quadratic assignment problem

Biological computing provides a promising approach to attacking computationally intractable problems. The quadratic assignment problem (QAP) is a well-known NP-hard combinatorial optimization problem. This paper addresses the problem of how to solve QAP under the Adleman–Lipton-sticker model. A theoretically efficient DNA algorithm for solving QAP is proposed, which is executed by performing O(Kn⁴) operations on test tubes of DNA molecular strands with n² + K + 1 bit regions, where n is the number of facilities, and K is the length of the binary representation of an upper bound on the objective function. With the rapid progress of molecular biology techniques, the proposed algorithm might be of practical use in treating medium-sized instances of QAP.     

The variable neighborhood search for the two machine flow shop problem with a passive prefetch

We consider the two machine flow shop scheduling problem with passive loading of the buffer on the second machine. To compute lower bounds for the global optimum, we present four integer linear programming formulations of the problem. Three local search methods with variable neighborhoods are developed for obtaining upper bounds. Some new large neighborhood is designed. Our methods use this neighborhood along with some other well-known neighborhoods. For computational experiments, we present a new class of test instances with known global optima. Computational results indicate a high efficiency of the proposed approach for the new class of instances as well as for other classes of instances of the problem.